WASHINGTON—At least two of the memos that former FBI Director James Comey gave to a friend outside of the government contained information that officials now consider classified, according to people familiar with the matter, prompting a review by the Justice Department’s internal watchdog.

Of those two memos, Mr. Comey himself redacted elements of one that he knew to be classified to protect secrets before he handed the documents over to his friend. He determined at the time that another memo contained no classified information,...